arXiv research

Study the limiting shape of solutions to the L_p-Minkowski problem as p approaches negative infinity.

Study flows for capillary Minkowski problems in half-spaces.

Paper solves capillary Lp-Minkowski problem for p>1.

The dual $L_p$-Minkowski problem with $p<0<q$ is investigated in this paper. By proving a new existence result of solutions and constructing an example, we obtain the non-uniqueness of solutions to this problem.

Sharp curvature estimates lead to optimal $C^{1,1}$ regularity for $L_p$ Minkowski problems.

Unique solution found for even L^p Minkowski problem at p=p0, but fails for p<p0.

Researchers solved the even $L^p$-Minkowski problem under curvature pinching.

This paper classifies solutions to a specific hyperbolic geometry problem.

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of…

Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.

Solves capillary curvature problems for specific p values.

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

This paper classifies the symmetry groups of a specific Monge-Ampère equation.

Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the …

Paper proves uniqueness of solutions to a geometric inequality problem.

Sharp inequality for eigenvalues of convex bodies, proving ellipsoid uniqueness.

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ i…

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère type equations subject to certain boundary conditions depending on the value of $p$. The special case of…

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature r…

Paper estimates diameter for Minkowski problem solutions.

Paper introduces new $L_p$ $q$-torsional measure and solves Minkowski problem.

Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.

We study the iterations of a class of curvature image operators $Λ_p^{\varphi}$ introduced by the author in (J. Funct. Anal. 271 (2016) 2133--2165). The fixed points of these operators are the solutions of the $L_p$ Minkowski problems with the positive continuous prescribed data $\varphi$. One of our results states tha…

Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, $L_p$-Minkowski and $L_p$-Brunn-Minkowski inequalities f…

Paper proves rigidity results for anisotropic capillary hypersurfaces.

Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.

This paper classifies solutions for a specific geometric problem.

Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.

Paper proves anisotropic Minkowski inequality and related inequalities.

Paper studies inverse curvature flows and solves related geometric problems.

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…

Optimal transport reformulates multiple quantile hedging problem.

Solves four problems related to circle families in the plane.

Solves four problems related to sphere families in 3D space.

The paper solves optimal control problems for various convex sets using convex trigonometry.

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we anal…

Study proves only origin-centered spheres solve certain curvature problems.

MathChat uses LLM agents to solve challenging math problems through conversational problem-solving.

The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.

Paper solves Gromov-Wasserstein for point clouds efficiently.

The paper explains how microlocal analysis solves geometric inverse problems.

Proves NP and co-NP status for knot core recognition in solid torus.

The min-max problem, also known as the saddle point problem, is a class of optimization problems which minimizes and maximizes two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal processing and communication (SPCOM) problems. Despite its popularity, most exist…

A new method solves complex control problems with random coefficients.

This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For ea…

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…